On self-avoiding polygons and walks: The snake method via pattern fluctuation
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY(2019)
摘要
For d >= 2 and n is an element of N, let W-n denote the uniform law on self-avoiding walks of length n beginning at the origin in the nearest-neighbour integer lattice Z(d), and write Gamma for a W-n-distributed walk. We show that in the closing probability W-n (parallel to Gamma(n)parallel to = 1) that Gamma's endpoint neighbours the origin and is at most n(-1/2+o(1)) in any dimension d >= 2. The method of proof is a reworking of that in [Ann. Probab. 44 (2016), pp. 955-983], which found a closing probability upper bound of n(-1/4+o(1)). A key element of the proof is made explicit and called the snake method. It is applied to prove the n(-1/2+o(1)) upper bound by means of a technique of Gaussian pattern fluctuation.
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